The PI will investigate the Galois cohomology groups of function fields of curves over number fields. Let F be the function field of a curve over a number field. If p is a prime and F contains a primitive pth root of unity, a symbol in degree n Galois cohomology is simply a cup product of elements in the degree one cohomology modulo p. Voevodsky's theorem implies that every element in the degree n Galois cohomology modulo p is a sum of symbols. It is proposed to study boundedness of symbol length for F for each prime p. This study has consequences for bounding the u-invariant of F. The PI also proposes to study questions concerning the admissibility of finite groups over function fields of curves over p-adic fields.
A basic pursuit of geometers and number theorist is to find common solutions of polynomial equations. A homogeneous polynomial of degree two is called a quadratic form. It is known that every quadratic form in at least five variables over a totally imaginary number field has non-trivial zero, thanks to the Hasse-Minkowski theorem. It is an open question whether every quadratic form in sufficiently many variables over the function field of a curve over a totally imaginary number field has a non-trivial zero. By a theorem of Voevodsky, quadratic forms are classified by Galois cohomology invariants over any field. We propose to study the existence of non-trivial zeros for quadratic forms in sufficiently many variables over function fields of curves over totally imaginary number fields, by studying boundedness questions for symbol lengths in Galois cohomology.
